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4.5 Digital Robot Control 225

(6)

for some positive definite P and Q.
Selecting now the Lyapunov function

(7)

(5) and (6) reveal
(8)

Assuming for simplicity that the disturbance d is zero, note that

(9)

is negative definite at each sample time. [If d is nonzero but bounded, then
V(e k) is negative outside some ball, and the discussion may be modified.]
The remainder of the proof follows from a theorem of [LaSalle and Lefschetz
1961] by showing that:

1.
2. V( (t 1))<V( (t 2)) for all t 2 t 1 0 when || (t 1)|| L and
|| (t 2 )||>L+r.

At issue is the fact (9) and the continuity of

This result shows that good tracking is obtained for all sample periods less
than some maximum sample period T M which depends on the specific robot
arm. A more refined result can show that the errors increase as the maximum
desired acceleration increases, or equivalently, that smaller sample periods
are required for larger desired accelerations.
We now discuss some issues in discretizing the inner nonlinear loop and
then the outer linear loop.

Discretization of Inner Nonlinear Loop
There is no convenient exact way to discretize nonlinear dynamics. Given
a nonlinear state-space system

(4.5.4)
Euler’s approximation yields

(4.5.5)

Copyright © 2004 by Marcel Dekker, Inc.